PSI - Issue 38

Emilien Baroux et al. / Procedia Structural Integrity 38 (2022) 497–506 E. Baroux et l. / Structural Integr ty P o edi 00 (2021) 1– ??

499

3

β ( M ) i

D M , σ ( t ) , t ∈ [0 , T ]] =

n c i

( ∆ τ )

(2)

S β

0 ( M )

2.3. Local stress on linear structures We make the simplifying assumption that our structure’s answer is quasi-static under service loads. It means that instantaneous local stress at each point of our structure depends only on instantaneous values of F . This quasi-static approximation is hindered by high frequency components such as pavements or large spectrum obstacles like sidewalks or potholes. Other fatigue characterization methods may take into account the e ff ects of high frequency, see for instance Benasciutti et al. (2013). We ignore the non-linear e ff ects of elements like rubber joints or mechanical stops, on the structure’s mechanical response. Local loads at each point of the structure are found using the principle of superposition

n F j

σ ( M , t ) =

K

F j ( t )

(3)

M , j

denoting K M , j stress localization second-order tensors associated to each global load component. We make the further assumption that the damage variable τ is a linear form τ = k f .σ. h f . Such a property holds for damage variables like shear stress calculated on point M ’s critical slip plane. From Eqs. 2 and 3, damage at point M is

  

1 0 ( M )    ∆

. h f F j )    β ( M ) i

   n F j

n c i

D ( M , F ) =

k f . K

(4)

S β

M , j

2.4. Pseudo-damage Choosing an adequate norm . , let us rewrite 1 S 0 ( M ) n F j k f . K M , j

A M e A M . F

. h f F j =

(5)

We propose to use the canonic 2-norm in space R n F . The term A M 2 is akin to stress magnification level around the point M in the structure. It depends on the structure’s rigidity, the point’s location, its geometry and its stress intensity factors. The unit vector e A M stems from privileged crack microscopic initiation and propagation directions. Several points in a car chassis may have the same stress orientation e A M . Thanks to the linearity of rainflow counting, we can define pseudo-damage ˆ D M ( F ) as

n c

∆ ( e A M . F )

i

β ( M ) i

β ( M ) 2

β ( M ) 2

ˆ D M ( e A M . F )

= A M

D ( M , F ) = A M

(6)

Eq. 6 is a generalization of pseudo-damage calculated from uniaxial load signals, presented in Johannesson (2014) Chap. 3. In the rest of the article, this exponent will be considered to be unvariably equal to 4 for the weak points to be designed. It is a conventional value for the Basquin exponent of weld beams’ fully reversed (R = -1) traction Wo¨hler curves. Therefore, we can rewrite ˆ D M = ˆ D . 2.5. Chassis weak points and damages Design experience helps to determine what kind of points one wants to design in a new car chassis. The structure directs stress directions around them. Local loading at these points can be associated to specific load cases, such as load cases presented in Fig. 1. Each load case’s induced damage is preponderant for di ff erent sensible points in the structure. We can deduce a satisfying pseudo-damage characterization of all validation points on a car chassis by picking a set of complementary load cases.